Most
Wanted Puzzle Solutions

"A
friend of mine beats me REPEATEDLY at Nim-game...
That frustrates me immensely! Are there some winning
algorithms you can apply to win at this game? Thanks
for your kind help!" -- Antoine

Category:
Mathematical game of strategy / Combinatorial gameName: Nim-game (or Marienbad-game,
aka tactix)Material: Matchsticks or any other counters.Aim of the game: The player who takes the last
counter loses.Editor's notice: The second player can always
win, that makes Nim a distinctly 'impartial game'.

SOLUTION

The
traditional Nim-game (aka Marienbad-game)
consists of four rows of 1, 3, 5 and 7 matchsticks
(or any other objects). Two players take any number
of matchsticks from one row alternately. The one,
who takes the last matchstick loses.

The
winning strategy is:
You must always take as many matchsticks as possible
so that the “Nim sum” of the rows remains
ZERO.

What
is a “Nim sum”?
Count the matchsticks in each row... And convert them
mentally in multiples of 4, 2 and 1. Then, CANCEL pairs
of equal multiples, and add what is left. So, when
starting, the “Nim sum” of the rows is:

Row1
= 1

=
1 x 1 = 1

=

1

Row2
= 3

=
1 x 2 + 1 x 1

=

2

1

Row3
= 5

=
1 x 4 + 1 x 1

= 4

1

Row4
= 7

=
1 x 4 + 1 x 2 +
1 x 1

= 4

2

1

Total
of UNPAIRED multiples

= 0

0

0

As
you can see, there are currently TWO 4’s, TWO
2’s, and FOUR 1’s (= TWO + TWO + FOUR
= 8). You have then an EVEN number of multiples,
the remainder after dividing this number (8) by 2
gives 0.

To
win at Nim-game, always make a move, whenever possible,
that leaves a configuration with a ZERO “Nim
sum”, that is with ZERO unpaired multiple(s)
of 4, 2 or 1. Otherwise, your opponent has the advantage,
and you have to depend on his/her committing an error
in order to win.

How
to leave a zero “Nim sum”:
Your opponent moves and leaves you the following configuration:

Row1
= 1

=
1 x 1

=

1

Row2
= 3

=
1 x 2 + 1 x 1

=

2

1

Row3
= 5

=
1 x 4 + 1 x 1

= 4

1

Row4
= 5

=
1 x 4 + 1 x 1

= 4

1

Total
of unpaired multiples

= 0

1

0

Get
rid of ONE 2, by taking 2 matchsticks
from the 2nd row. That leaves your opponent at 1,
1, 5, 5 which is, for him, a losing configuration...

Row1
= 1

=
1 x 1

=

1

Row2
= 1

=
1 x 1

=

1

Row3
= 5

=
1 x 4 + 1 x 1

= 4

1

Row4
= 5

=
1 x 4 + 1 x 1

= 4

1

Total
of unpaired multiples

= 0

0

0

Every
time you leave your opponent a zero “Nim sum” configuration,
you increase your chances to win! Here below are
listed all the possible zero “Nim sum” configurations
(sometimes, the order has no importance, for example:
3, 3, 1, 1 or 3, 1, 3, 1 has the same result). You
can print the table and use it as a cheat sheet...
But you can also improve your concentration skills
by practicing “Nim sums” mentally!

Final
considerations
As you can see, the starting configuration 1, 3, 5,
7 is a losing one for the player who starts the game.
Furthermore, the player who starts the game first will
obviously lose if his opponent takes care to keep during
the whole game play a zero “Nim sum” configuration.
In conclusion, it is ALWAYS disadvantageous
to start the game first!

Behind
the challenge
You have learned that the key to win at Nim-game is
the binary digital sum (Nim sum). This operation is
also known as 'exclusive or' (xor) addition. Your scientific
calculator can be of help to find any “Nim sum” of
two or more integers thanks to its XOR operator button.
To 'xor' or 'Nim-add' several numbers, just type into
your calculator the first number then press the XOR
button and type the following number to 'Nim-add',
and so on... When you have entered all the numbers,
press the equal sign, as follows:
1 xor 3 xor 5 xor 7 = 0

How
does a XOR
logic operator work?The “Nim sum” of two (or more)
integers is calculated by means of ‘xor’ bitwise
operator (symbol ⊕). In logic,axor b =
[(a or b) and (not (a and b))]
In other words, a ‘xor’ bitwise
operation returns a 1 in a bit position if bits
of one but not both operands are 1’s. For instance:
1012 -> 510 number
A
1112 -> 710 number
B
----
0102 -> 210 The “Nim
sum” of numbers A and B, 5 ⊕ 7 = 2

XOR
logic operator in the graphic design worldGraphic
designers who work with vector-based softwares use
a pathfinder feature, similar to the XOR operator,
to get rid of everything is overlapped. For example,
when 2 shapes are overlapped as shown in fig. a,
the XOR button in the Adobe
Illustrator pathfinder tool (called “exclude
overlapping shape areas” button, see fig. below)
allows to exclude any shape areas that were overlapping
at all, like illustrated in fig. b.